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Submitted on 21 Feb 2018
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Strong convergence rates of semi-discrete splitting
approximations for stochastic Allen–Cahn equation
Charles-Edouard Bréhier, Jianbo Cui, Jialin Hong
To cite this version:
approximations for stochastic Allen–Cahn equation
Charles-Edouard Br´
ehier, Jianbo Cui, and Jialin Hong
Abstract. This article analyzes an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise, in a bounded spatial domain with smooth boundary in dimension d ≤ 3. The splitting strat-egy is combined with an exponential Euler scheme of an auxiliary problem.
When d = 1 and the driving noise is a space-time white noise, we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity, we then prove that under very mild assumptions on the initial data, this scheme achieves the optimal strong convergence rate O(δt14). When d ≤ 3 and the driving noise possesses some regularity in space, we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension d = 1, these properties are used to prove that the splitting scheme has a strong convergence rate O(δt).
1. Introduction
The stochastic Allen-Cahn equation driven by an additional noise term models the effect of thermal perturbations, and plays an important role in the phase theory and the simulations of rare events in infinite dimensional stochastic systems (see e.g. [13, 19, 27]).
In this article, we mainly focus on deriving the optimal strong convergence rates of temporal splitting schemes for the stochastic Allen-Cahn equation driven by Wiener processes, including the cylindrical Wiener process and some more regular Wiener processes, under homogenous Dirichlet boundary conditions:
dX(t) = AX(t) + F (X(t))dt + dWQ(t), t ∈ (0, T ], X(0) = X0,
(1)
where F (x) = x − x3, (WQ(t))
t∈[0,T ] is a generalized Wiener process on a filtered
probability space (Ω,F , (F (t))t∈[0,T ], P) and O ∈ Rd, d ≤ 3 is a bounded spatial
domain with smooth boundary ∂O.
Strong convergence of numerical approximations for Stochastic Partial Differ-ential Equations(SPDEs) with globally Lipschitz continuous coefficients has been extensively studied in the last twenty years (see e.g. [1, 14, 17, 21, 25]). For SPDEs with non-Lipschitz coefficients, there only exist a few results about the strong convergence rates of numerical schemes (see e.g. [2, 3, 8, 9, 12]). The
2010 Mathematics Subject Classification. Primary 60H35; Secondary 60H15, 65M15. Key words and phrases. Stochastic Allen-Cahn equation, splitting scheme, strong conver-gence rate, exponential integrability.
strong convergence rates of numerical schemes, especially the temporal discretiza-tion, is far from being understood and it is still an open problem to derive general strong convergence rates of numerical schemes for SPDEs with non-globally Lips-chitz coefficients.
For the discretization of equations such as the stochastic Allen-Cahn equation, the main difficulty is the polynomial growth of the non-globally Lipschitz continuous coefficient F . It is very delicate and necessary to design efficient numerical schemes for stochastic equations with this type of nonlinearities. The authors in [20] study a fully implicit split-step scheme combined with the backward Euler scheme, and show that the scheme converges strongly with a rateO(δt12) for Eq. (1) with d ≤ 3, driven by some Q-Wiener processes. We refer to [12] for the analysis of finite element methods applied to stochastic Allen-Cahn equations with multiplicative noise. For Eq. (1), with d = 1 driven by a space-time white noise, first, the authors in [3] obtain the strong convergence rate results for a nonlinearity-truncated Euler-type scheme. Similar strong convergence results are then obtained in [2] for a nonlinearity-truncated fully discrete scheme. A backward Euler-Spectral Galerkin method has been considered in [22] by using stochastic calculus in martingale type 2 Banach spaces. Recently, the authors in [4] propose some splitting schemes and prove the proposed schemes are strongly convergent without strong convergence rates.
In this work, we give a systemic analysis of the properties of a splitting scheme and its strong convergence rates for approximating Eq. (1) with d ≤ 3 driven by different kinds of noise. We first introduce the splitting scheme with a time-step size δt > 0, defined by:
Yn = Φδt(Xn), (2) Xn+1= SδtYn+ Z tn+1 tn S(tn+1− s)dWQ(s), where Φδt(z) = √ z
z2+(1−z2)e−2δt is the phase flow of dX = F (X(t))dt, t ∈ [0, δt], X0= z, and Sδt = S(δt) = eAδt. This type of splitting scheme, in a stochastic context,
has been first proposed in [4], and it is convenient for practical implementations since it is explicit and strongly convergent without a taming or truncation strategy. Note that an exponential Euler scheme is used in the second step of the splitting strategy.
In this article, we first derive the optimal strong convergence rate of the split-ting scheme in the case of space-white time noise, using a variational approach. This gives a positive answer to the question asked in [4], concerning the strong convergence rate of splitting schemes for the stochastic Allen-Cahn equation. We would like to mention that these splitting-up based methods have many applica-tions on approximating SPDEs with the Lipschitz nonlinearity, and are also used for approximating SPDEs with non-Lipschitz or non-monotone nonlinearities (see e.g. [7, 9, 11, 15]).
In order to analyze the strong convergence rate of this splitting method for different types of noise, different approaches are required. In the case of space-time white noise, there are three main steps to derive the strong convergence rate. Following [4], the first step is constructing an auxiliary problem, with a modified nonlinearity Ψδt instead of F , such that the splitting scheme can be viewed as a
the exponential Euler applied to the original equation may be divergent, the so-lutions of the numerical scheme and of the auxiliary problem are proved to be bounded in Lp(Ω; Lq), for all finite p, q. Thus no taming or truncation strategy is required to ensure the boundedness of numerical solutions. The second step is based on the monotonicity properties of the nonlinearities F and Ψδt, appearing
in the exact and auxiliary problems respectively. In addition, since the noise is additive and an exponential Euler scheme is used with no discretization of the sto-chastic convolution, one is lead to study some PDEs with random coefficients. The last step consists in applying properties of the stochastic convolution and stochastic calculus results in martingale type 2 Banach spaces, to deduce the optimal strong convergence rateO(δt14) in Lp(Ω; C(0, T ; Lq)), p ≥ q = 2m, m ∈ N+, i.e.,
sup t∈[0,T ] kXN(t) − X(t)k Lq Lp(Ω)≤ C(X0, T, p, q)δt 1 4.
This variational approach can also be used to obtain the strong convergence rate O(δt1
2), in the case of more regular Q-Wiener processes, in dimension d ≤ 3. In the case of H1-valued Q-Wiener processes, in dimension d = 1, we get
higher strong convergence rates of this splitting method, thanks to exponential integrability properties of the exact and numerical solutions. To the best of our knowledge,this is the first result with strong convergence order 1 about the temporal numerical schemes approximating the stochastic Allen–Cahn equation. For similar approaches to derive the strong convergence rates of numerical schemes, we refer to [7, 16, 18] and the references therein. We first study stability and exponential integrability properties of the exact solution, in dimensions d ≤ 3, and obtain results of their own interest beyond analysis of numerical schemes. Then, in dimension d = 1, a new auxiliary processes ZN is constructed, and some a priori estimate and exponential integrability properties of ZN are studied. We then prove that the
scheme, in this context, has strong convergence order equal to 1:
sup n≤N kX N(t n) − X(tn)kLq Lp(Ω)≤ C(X0, Q, T, p, q)δt.
This strong convergence result is restricted to dimension d = 1, due to a loss of exponential integrability of the auxiliary process ZN in higher dimension. Further
study is required to overcome this issue.
This article is organized as follows. Some preliminaries are given in Section 2. The variational approach to deal with the case of space-time white noise and Q-Wiener processes, as well as some properties of the auxiliary problem, and one main strong convergence rate result, are given in Section 3. In Section 4, stability and exponential integrability properties, of the exact solution and of a new auxiliary processes, are studied. Finally, we establish the optimal strong convergence rate of this proposed scheme in dimension 1.
We use C to denote a generic constant, independent of the time step size δt, which differs from one place to another.
2. Preliminaries
In this section, we first introduce some useful notations and further assump-tions. Let T > 0, δt is the time step size, N is the positive integer such that N δt = T , and let {tk}k≤N be the grid points, defined by tk = kδt. We denote by
operator, which generates an analytic and contraction C0-semigroup S(t), t ≥ 0 in H
and Lq. It is well-known that the assumptions on O implies that the existence of the eigensystem {λk, ek}k∈N+ of H, such that λk > 0, −Aek= λkek and lim
k→∞λk = ∞.
Let Wr,q is the Banach space equipped with the norm k · k
Wr,q := k(−A)
r
2 · kLq for the fractional power (−A)r2, r ≥ 0. The identities H1= H1
0 and H2= H01∩ H2are
frequently used in Section 4.
Given two separable Hilbert spaces H and eH, we denote by L02(H, eH) the space of Hilbert-Schmidt operators from H into eH, equipped with the usual norm given by kQkL0 2(H, eH) = (P k∈N+kQekk2 e H) 1
2, where N+= {1, 2, · · · }, and the result does not depend on the orthonormal basis {ek}k∈N+ of H. We denote by Ls2 := L02(H, Hs), for s ∈ N.
Given a Banach space E, we denote by R( eH, E) the space of γ-radonifying operators endowed with the norm defined by kQkγ( eH,E)= (eEkPk∈N+γkQekk2E)
1 2, where (γk)k∈N+is a sequence of independent N (0, 1)-random variables on a proba-bility space (eΩ, fF ,P). We also need the Burkerholder inequality in martingale-typee 2 Banach spaces E = Lq, q ∈ [2, ∞), (see e.g. [5, 26]): for some Cp,E ∈ (0, ∞),
(3) sup t∈[0,T ] Z t 0 φ(r)dW (r) E Lp(Ω)≤ Cp,EkφkLp(Ω;L2([0,T ];γ( eH;E))) = Cp,E E Z T 0 kφ(t)k2 γ( eH;E)dt p21p
and the following property (see [26]): for some Cq ∈ (0, ∞),
kφk2γ( eH,Lq)≤ Cq X k∈N+ (φek)2 Lq2, φ ∈ γ( eH, L q). (4) The process W :=P
k∈N+βkek is the H-valued cylindrical Wiener process, where
(βk)k∈N+are independent Brownian motions defined on a filtered probability space (Ω,F , (F (t))t∈[0,T ], P). The driving noise is WQ := PkβkQek, where Q is a
bounded operator from H to E. When Q = I, E = H, WQ is the standard cylindrical Wiener process, which corresponds to the case of space-time white noise. In Sections 3 and 4, we will also consider more regular cases, with assumptions Q ∈ Ls2, s ∈ N.
The solution of the stochastic Allen-Cahn equation, Eq. (1), is interpreted in a mild sense, X(t) = S(t)X0+ Z t 0 S(t − s)F (X(s))ds + Z t 0 S(t − s)dWQ(s). (5)
Let ω(t) =R0tS(t − s)dWQ(s) be the so-called stochastic convolution. Then note that Y (t) = X(t) − ω(t) solves a random PDE (written in mild form):
Y (t) = S(t)X0+
Z t
0
S(t − s)F (Y (s) + ω(s))ds. (6)
as follows: Xn+1= SδtΦδt(Xn) + Z tn+1 tn S(tn+1− s)dWQ(s) = SδtXn+ δtSδtΨδt(Xn) + Z tn+1 tn S(tn+1− s)dWQ(s),
where Ψδt(z) = Φδt(z)−zδt , Ψ0(z) = F (z), Φ0(z) = z. Thus we get for all n ∈
{0, . . . , N } Xn= S(tn)X0+ δt n−1 X k=0 S(tn+1− tk)Ψδt(Xk) + n−1 X k=0 Z tk+1 tk S(tn+1− s)dWQ(s).
A continuous time interpolation, such that XN(tn) = Xn for all n ∈ {0, . . . , N }, is
defined by XN(t) = S(t)X0+ Z t 0 S((t − bscδt))Ψδt(XN(bscδt))ds + Z t 0 S(t − s)dWQ(s), (7) where bscδt= max{0, δt, 2δt, · · · } ∩ [0, s].
As observed in [4], the proposed splitting scheme can be viewed as the expo-nential Euler method applied to the following auxiliary SPDE:
dXδt(t) = AXδt(t)dt + Ψδt(Xδt(t))dt + dWQ(t), Xδt(0) = X0.
(8)
The associated mild formulation is given by
Xδt(t) = S(t)X0+ Z t 0 S(t − s)Ψδt(Xδt(s))ds + Z t 0 S(t − s)dWQ(s).
Let Yδt(t) = Xδt(t) − ω(t), where ω is the stochastic convolution. Then Yδtis also
solution of a random PDE:
Yδt(t) = S(t)X0+
Z t
0
S(t − s)Ψδt(Yδt(s) + ω(s))ds.
(9)
We quote the following results from [4]. The estimates may be derived with elementary calculations.
Lemma 2.1. For every δt0∈ (0, 1) and δt ∈ [0, δt0), the mapping Φδt is
glob-ally Lipschitz continuous, and the mapping Ψδt is locally Lipschitz continuous and
satisfies a one-side Lipschitz condition. More precisely, for q = 2m, m ∈ N+, |Φδt(z1) − Φδt(z2)| ≤ eCδt0|z1− z2|,
(Ψδt(z1) − Ψδt(z2))(z1− z2)q−1≤ eCδt0|z1− z2|q,
|Ψδt(z1) − Ψδt(z2)| ≤ C(δt0)|z1− z2|(1 + |z1|2+ |z2|2),
|Ψδt(z1) − Ψ0(z1)| ≤ C(δt0)δt(1 + |z1|5).
3. Strong convergence rate analysis of the splitting scheme approximating stochastic Allen-Cahn equation by a variational
approach
includes the cases of the cylindrical Wiener process (Q = I, q = 1) and of Lq -valued Q-Wiener processes (Q ∈ γ(H, Lq)).
We recall that in [4] it is proved that the scheme is convergent, when d = 1 and Q = I. Precisely, assume that X0∈ Hβ1∩ E, for some β1> 0. Then
lim δt→0E h sup n≤N kXN(tn) − X(tn)kp i = 0.
However, it is well-known that the standard approach used to derive strong rates of convergence using a Gronwall’s inequality argument, cannot be applied, when the nonlinearity is not globally Lipschitz continuous. Additional properties, precisely giving exponential integrability for the exact and numerical solutions, are required. Instead, in the present section, we overcome this issue using a variational ap-proach, based on a different decomposition of the error introduced below.
For convenience, throughout this article, we assume that X0 is a
determinis-tic function and that supk∈N+kekkE ≤ C. The typical example to ensures that
supk∈N+kekkE ≤ C is the d dimensional cube [0, 1]d.
3.1. A priori estimates and spatial regularity properties. We first deal with the case Q = I, d = 1 and recall the following well-known result about the stochastic convolution (see e.g. [10]): for 2 ≤ q < ∞,
E h sup t∈[0,T ] kω(t)kpLq i ≤ Cp(T ), E h sup t∈[0,T ] kω(t)kpEi≤ Cp(T ) < ∞.
The following lemma states standard a priori estimates for the processes X, XN and Yδt defined by Eq. (1), (7) and (9) respectively. For convenience, throughout this paper, we omit the mollification procedure to get the evolution of k · kLq.
Lemma 3.1. Let d = 1, Q = I, q = 2m, m ∈ N+, p ≥ 1 and X0 ∈ Lq. Then
X,Yδt and XN satisfy
E h sup t∈[0,T ] kX(t)kpLq i < C(T, p, q)(1 + kX0kpLq), and E h sup t∈[0,T ] kYδt(t)kp Lq i + Eh sup t∈[0,T ] kXN(t)kp Lq i < C(T, p, q)(1 + kX0kpLq).
Proof. For the a priori estimate for the exact solution X, we refer to [10]. Thus we focus on the a priori estimate of Yδt and XN. The definition, Eq. 9, of
Yδt and the one-side Lipschitz condition on Ψ
H¨older and Young inequalities, imply that for 2 ≤ q < ∞, kYδt(t)kqLq ≤ kX0k q Lq+ q Z t 0 hAYδt(s), (Yδt(s))q−2Yδt(s)ids + q Z t 0 hΨδt(Yδt(s) + ω(s)), (Yδt(s))q−2Yδt(s)ids ≤ kX0kqLq+ q Z t 0 hΨδt(Yδt(s) + ω(s)) − Ψδt(ω(s)), (Yδt(s))q−2Yδt(s)ids + q Z t 0 hΨδt(ω(s)), (Yδt(s))q−2Yδt(s)ids ≤ kX0k q Lq+ C(δt0, q) Z t 0 kYδt(s)kq Lqds + C(δt0, q) Z t 0 kΨδt(ω(s))kLqkYδt(s))kq−1Lq ds ≤ kX0k q Lq+ C(δt0, q) Z t 0 kYδt(s)kqLqds + C(δt0, q) Z t 0 (1 + kω(s)k3qL3q)ds. Using the moment estimate on the stochastic convolution above, applying the Gron-wall’s inequality concludes the proof for Yδt.
The estimate for XN is proved using similar arguments. First, note that it is
sufficient to control the values of XN at the grid points t
n, n ≤ N : E h sup t∈[0,T ] kXN(t)kp Lq i ≤ C(p, q, T )Ehsup n≤N kXN(t n)k p Lq i . By the definition of XN(t
n) = Xn, n ≤ N and the Lipschitz continuity of Φδt
stated in Lemma 2.1, since S(t) is a contraction semigroup, we obtain
kXN(t n)−ω(tn)kLq ≤ S(δt)Φδt(X N(t n−1)) − S(δt)ω(tn−1) Lq ≤ Φδt(X N(t n−1)) − Φδt(ω(tn−1)) Lq+ Φδt(ω(tn−1)) − ω(tn−1) Lq ≤ eCδt X N(t n−1) − ω(tn−1) Lq+ Cδt(1 + kω(tn−1)k 3 L3q).
Then using the discrete Gronwall’s inequality, and the estimate on the stochastic convolution, we get E h sup n≤N kXN(tn)k p Lq i ≤ C(T, q, p), which concludes the proof of XN.
We now study spatial regularity properties of the processes XN and X. We
first state a Lemma (see [10]) concerning the factorization method.
Lemma 3.2. Assume that p > 1, r ≥ 0, γ > 1p + r and that E1 and E2 are
Banach spaces such that
if f ∈ Lp([0, T ]; E2).
Lemma 3.3. Assume that d = 1, Q = I, p ≥ 2 and kX0kHβ1 < ∞, β1> 0. The
solution u satisfy the following estimate: if β < min(12, β1), then
E h sup 0≤s≤T kX(t)kp Hβ i ≤ C(p, T, β, X0) < ∞.
Proof. It is known (see e.g. [10]) that, for β < 12,
E h sup 0≤s≤T kω(s)kp Hβ i ≤ C(p, T ).
Thus we only need to study the regularity of S(t)X0 and of the deterministic
convolutionR0tS(t − s)F (X(s))ds. First,
kS(t)X0kHβ1 ≤ kX0kHβ1.
For the deterministic convolution, by the Fubini theorem, we have
Z t 0 S(t − s)F (X(s))ds = sin γπ π Z t 0 (t − s)γ−1S(t − s)Yγ(s)ds, Yγ(t) = Z t 0 (t − s)−γS(t − s)F (X(s))ds where we choose γ < 1
4 such that the regularity result also holds for the stochastic
convolution. Notice that kS(t)xkHβ ≤ M t− β 2kxk
H, for β > 0. Taking E1= Hβ and
E2= H, r = β2. Lemma 3.2 yields that for large enough p and γ > β2 +1p,
E h sup 0≤t≤T Z t 0 S(t − s)F (X(s))ds p Hβ i ≤ CEh Z T 0 Yγ(t) p H dti ≤ CEh Z T 0 t−2γkS(t)kL(H,H)1 + sup r∈[0,T ] kX(r)k3 L6 dt pi ≤ C(T, p, X0).
This concludes the proof.
Using standard arguments, including the use of a discrete Gronwall’s lemma, and the two lemmas stated above, one may derive the following almost sure result (see [4] for similar arguments): assume d = 1, Q = I, β < 12, X0∈ Hβ1∩ E, β1> 0.
Then almost surely, for some C(ω) ∈ (0, ∞), one has
sup n≤N kXN(t n) − X(tn)k ≤ C(ω)δtmin( β 2,β12 ).
3.2. Optimal strong convergence rate in space-time white noise case. We are now in position to apply the variational approach developed in [3] in order to obtain strong convergence rates for the splitting scheme (2).
We first state the main result of this section.
Theorem 3.1. Assume that d = 1, Q = I, kX0kL9q < ∞, p ≥ q = 2m, m ∈ N+ and η < 1 q. Then X N satisfies sup t∈[0,T ] kXN(t) − X(t)k Lq Lp(Ω)≤ C(T, X0, p, q)δt min(1 4,η), If in addition kX0kWβ,3q < ∞, β > 0, then sup t∈[0,T ] kXN(t) − X(t)kLq Lp(Ω)≤ C(T, X0, p, q)δt min(1 4, β 2+η),
Note that, for q ∈ [2, 4), the first estimate of Theorem 3.1 gives order of conver-gence 14. If q ∈ [4, ∞),the order of convergence 14 is obtained thanks to the second estimate, under the assumption β > 1
2− 2 q.
Observe that the error can be decomposed as follows:
kXN(t) − X(t)k
Lq ≤ kXN(t) − ω(t) − Yδt(t)kLq+ kYδt(t) + ω(t) − X(t)kLq ≤ kXN(t) − Xδt(t)k
Lq+ kYδt(t) − Y (t)kLq.
Then Theorem 3.1 is a straightforward consequence of the two auxiliary results stated below.
Proposition 3.1. Assume that d = 1, Q = I, kX0kL5q < ∞. Then the proposed method XN is strongly convergent to X and satisfies
sup t∈[0,T ] kYδt(t) − Y (t)k Lq Lp(Ω)≤ C(T, X0, p, q)δt, where p ≥ q = 2m, m ∈ N+.
Note that Yδt(t) − Y (t) = Xδt(t) − X(t), and in the case q = 2, Proposition 3.1 has already been proved in [4].
Proposition 3.2. Assume that d = 1, Q = I, kX0kL9q < ∞, p ≥ q = 2m, m ∈ N+. Then the proposed method XN satisfies for η <1
q, sup t∈[0,T ] kXN(t) − Xδt(t)kLq Lp(Ω)≤ C(T, X0, p, q)δt min(1 4,η). (10)
If in addition assume that kX0kWβ,3q< ∞, β > 0, then for η < 1q, sup t∈[0,T ] kXN(t) − Xδt(t)kLq Lp(Ω)≤ C(T, X0, p, q)δt min(1 4,η+ β 2). (11)
Proof of Proposition 3.1. Note that Yδt(0) = Y (0), and recall that Yδt(t)− Y (t) = Xδt(t) − X(t). Using the differential forms of the random PDEs (6) and (9),
kYδt(t) − Y (t)kq Lq = q Z t h(Yδt(s) − Y (s))q−2(Yδt(s) − Y (s)), AYδt(s) − AY (s)ids + q Z t h(Xδt(s) − X(s))q−2(Xδt(s) − X(s)), Ψ δt(X(s)) − Ψδt(X(s)))ids + q Z t h(Xδt(s) − X(s))q−2(Xδt(s) − X(s)), Ψ δt(X(s))) − F (X(s)))ids.
Thanks to Lemma 2.1, combined with Young’s inequality and Gronwall’s lemma, we obtain kYδt(t) − Y (t)kqLq ≤ C(T )δtqZ T 0 1 + Y (s) + ω(s) 5q L5q ds,
It remains to use the a priori estimates of Lemma 3.1 to conclude the proof. To prove Proposition 3.2, we follow the approach from [3], and we introduce an additional auxiliary process,
b
Y (t) := S(t)X0+
Z t
0
S(t − s)Ψδt(XN(bscδt))ds
for which the following auxiliary result is satisfied.
Lemma 3.4. Assume that d = 1, Q = I, kX0kL3q < ∞, p ≥ q ≥ 2. Then for 0 < η < 1, s ≥ δt, E h Y (s) + ω(s) − Xb N(bsc δt) p Lq i ≤ C(T, p, η, X0)(1 + (bscδt)−ηp)(δt)min( 1 4,η)p. (12)
If assume in addition that kX0kWβ,q< ∞, β > 0, then we have
E h Y (s) + ω(s) − Xb N(bsc δt) p Lq i ≤ C(T, p, η, X0)(1 + (bscδt)−ηp)(δt)min( 1 4, β 2+η)p. (13)
Thanks to the smoothing properties of the semigroup S(t), we have for arbitrary η < 1, I1≤ S(bscδt)(S(s − bscδt) − I)X0 Lq ≤ C A ηS(bsc δt) L(Lq,Lq) A −η(S(s − bsc δt) − I)X0 Lq ≤ C(bscδt)−ηδtηkX0kLq.
Then we turn to estimate the term I2, for s ≥ ,
I2≤ Z bscδt 0 S(s − r) − S(bscδt− brcδt) Ψδt(XN(bscδt))dr Lq + Z s bscδt S(s − r)Ψδt(XN(bscδt))dr Lq ≤ Z bscδt 0 S(s − r)(S(r − brcδt) − I)Ψδt(X N(bsc δt)) Lqdr + Z bscδt 0 S(bscδt− brcδt)(S(s − bscδt) − I)Ψδt(X N(bsc δt)) Lqdr + Cδt sup s∈[0,T ] Ψδt(X N(bsc δt)) Lq.
Similar to the estimate of I1, combing with Lemma 2.1, we obtain for 0 < η < 1,
I2≤ C(T )(δtη+ δt)(1 + sup n≤N
kXN(t
n)k3L3q). Thanks to the Burkholder inequality, Eq. (3), from Section 2.1,
E[I3p] ≤ C(p)E h Z bscδt 0 S(s − r) − S(bscδt− r)dW (r) p Lq i + C(p)Eh Z s bscδt S(s − r)dW (r) p Lq i ≤ C(p) Z bscδt 0 S(s − r) − S(bscδt− r) 2 γ(H,Lq)dr p2 + C(p) Z s bscδt S(s − r) 2 γ(H,Lq)dr p2 . Thanks to Eq. (4), kφk2γ(H,Lq)≤ Cq X k∈N+ (φek)2 Lq2, φ ∈ γ(H, L q ).
Recall that it is assumed that sup
k∈N+
kekkE ≤ C < ∞. Moreover, one has the
following useful inequality: for any γ > 0 and any α ∈ [0, 1],
sup r∈(0,∞) X k∈N+ r12+αλα ke−γλk r= C(γ, α) < ∞.
E[I3p] ≤ C Z bscδt 0 X k∈N+ S(bscδt− r) S(s − bscδt) − Iek 2 Lqdr p2 + C Z s bscδt X k∈N+ kS(s − r)ekk2Lqdr p2 ≤ C Z bscδt 0 X k∈N+ e−2λk(bscδt−r)(e−λk(bscδt−s)− 1)2dr p2 + C Z s bscδt X k∈N+ e−2λk(s−r)dr p 2 ≤ C Z bscδt 0 X k∈N+ e−2λk(bscδt−r)λ 1 2 kδt 1 2dr p2 + C Z s bscδt (s − r)−12dr p2 ≤ C Z bscδt 0 r−34drδt 1 2 p2 + Cδtp4 ≤ Cδt p 4.
Combining the estimates of I1, I2 and I3, we obtain for s ≥ δt,
E h Y (s) + ω(s) − Xb N(bsc δt) p Lq i ≤ C(p)E[I1p] + E[I p 2] + E[I p 3] ≤ C(T, p, q, X0) (bscδt)−ηpδtηp+ δt p 4 ≤ C(T, p, q, X0)(1 + (bscδt)−ηp)δtmin( 1 4,η)p, which shows the first assertion.
If in addition we have kX0kWβ,q < ∞, β > 0, alternatively we have I1≤ A ηS(bsc δt)kL(Lq,Lq)kA−η(S(s − bscδt) − I)X0 Lq ≤ C(bscδt)−ηδtη+ β 2kX0k Wβ,q,
where η +β2 ≤ 1, 0 < η < 1. Combing the previous estimation on I2 and I3, this
concludes the proof.
It now remains to prove Proposition 3.2, using Lemma 3.4.
Proof of Proposition 3.2. We first show the estimation (10) with the rough initial datum X0∈ L9q. Due to the definition of XN and Yδt, we have
The first term is controlled by the smoothing properties of S(t) and the uniformly boundedness of Ψδt(XN(bscδt)). For 0 < η1< 1, we have
Z t 0 S(t − bscδt)Ψδt(XN(bscδt))ds − Z t 0 S(t − s)Ψδt(XN(bscδt))ds Lq = Z t 0 Aη1S(t − s)A−η1(S(s − bsc δt) − I)Ψδt(XN(bscδt))ds Lq ≤ Z t 0 C(t − s)−η1(s − bsc δt)η1 Ψδt(X N (bscδt)) Lqds ≤ C(η)δtη1 Z t 0 (1 + kXN(bscδt)k3L3q)ds.
We use the auxiliary process bY and (12) in Lemma 3.4 to deal with the second term since Z t 0 S(t−s)Ψδt(XN(bscδt))ds− Z t 0 S(t−s)Ψδt(Yδt(s)+ω(s))ds Lq= kY δt(t)− bY (t)k Lq. By the one-sided Lipschitz continuity of Ψδt, H¨older and Young inequality, we have
for δt ≤ t, kYδt(t) − bY (t)kq Lq = kYδt(δt) − bY (δt)kqLq+ q Z t δt h(Yδt(s) − bY (s))q−2(Yδt(s) − bY (s)), AYδt(s) − A bY (s)ids + q Z t δt h(Yδt(s) − bY (s))q−2(Yδt(s) − bY (s)), Ψ δt(Yδt(s) + ω(s)) − Ψδt(XN(bscδt))ids ≤ kYδt(δt) − bY (δt)kqLq+ q Z t δt D (Yδt(s) − bY (s))q−2(Yδt(s) − bY (s)), Ψδt(Yδt(s) + ω(s)) − Ψδt( bY (s) + ω(s)) E ds + q Z t δt h(Yδt(s) − bY (s))q−2(Yδt(s) − bY (s)), Ψ δt( bY (s) + ω(s)) − Ψδt(XN(bscδt))ids ≤ kYδt(δt) − bY (δt)kq+ C(q) Z t δt kYδt(s) − bY (s)kq Lqds + Z t δt Ψδt( bY (s) + ω(s)) − Ψδt(X N(bsc δt)) q Lqds. Then Gronwall’s inequality yields that for t ≥ δt,
kYδt(t) − bY (t)kqLq ≤ e CT kYδt(δt) − bY (δt)kqLq + eCT Z T δt kΨδt( bY (s) + ω(s)) − Ψδt(XN(bscδt))k q Lqds.
Using Lemma 2.1 and H¨older inequality,
leads that kYδt(t) − bY (t)k Lq≤ C(T, q) kYδt(δt) − bY (δt)kLq+ sup s∈[0,T ] 1 + k bY (s)k2L3q+ kω(s)k2L3q + kXN(bscδt)k2L3q Z T δt Y (s) + ω(s) − Xb N(bsc δt) q L3q 1q ds ! . Since for t ≤ δt, sup t∈[0,δt] kYδt(t) − bY (t)k Lq ≤ C Z t 0 S(t − s)Ψδt(Yδt(s) + ω(s))ds Lq + C Z t 0 S(t − s)Ψδt(XN(bscδt))ds Lq ≤ Cδt sup s∈[0,T ] 1 + kYδt(s)k3L3q+ kω(s)k3L3q+ kXN(s)k3L3q .
Taking expectation, together with the above results and a priori estimate in Lemma 3.1, H¨older inequality and Minkowski’s inequality, leads that, for p ≥ q, η < 1q,
By Minkowski’s inequality and (12) in Lemma 3.4, we get for η < 1q, Z T δt Y (s) + ω(s) − Xb N(bsc δt) q L3qds 1q L2p(Ω) ≤ Z T δt E h Y (s) + ω(s) − Xb N(bsc δt) 2p L3q i2pq ds 1 q ≤ C1 + Z T δt bsc−ηqδt ds 1 q δtmin(14,η)≤ C(T, p, q, kX0k L9q)δtmin( 1 4,η), which combing with the above estimation, yields that
E h sup t∈[δt,T ] kYδt(t) − bY (t)kp Lq i ≤ C(T, p, q, kX0kL9q)δtmin( 1 4,η).
By the continuity of Yδt(t) and bY (t), together the above estimations, we have for
E h sup t∈[0,T ] kYδt(t) − bY (t)kp Lq i ≤ Eh sup t∈[δt,T ] kYδt(t) − bY (t)kp Lq i + Eh sup t∈[0,δt] kYδt(t) − bY (t)|p Lq i ≤ C(T, p, q, kX0kL9q)δtmin( 1 4,η),
which establishes the first assertion (10). For the estimation (11), we use (13) to estimate the term Ehsupt∈[0,T ]kYδt(t) − bY (t)k
p Lq
i
and the arguments are similar. By this variational approach, we can deduce that if d = 1, Q = I, p ≥ q = 2m, m ∈ N+, β > 0, η < 1q, X0 ∈ Wβ,q∩ E. Then the strong convergence rate result
still holds, i.e., sup t∈[0,T ] kXN(t) − X(t)k Lq Lp(Ω)≤ C(T, p, q, X0)δt min(1 4, β 2+η), by using the estimation
kΨδt(z1) − Ψδt(z2)kLq ≤ Ckz1− z2kLq(1 + kz1k2E + kz2k2E)
and the procedures of Theorem 3.1. This above result gives the answer to the problem about the strong convergence rates of splitting schemes appeared in [4].
Remark 3.1. This above variational approach, combining with some further analysis on the discrete stochastic convolution, may also be available for obtaining the optimal strong convergence rates of other splitting schemes, such as the split-ting exponential Euler scheme and the splitsplit-ting implicit Euler scheme in [4]. This extension will be studied in future works.
To conclude this section, we give extensions of Theorem 3.1, when Eq. (1) is driven by a Q-Wiener process, in dimension d ≤ 3. We only sketch the proofs of the parts which require nontrivial modifications. Note that the order of convergence depends on the H¨older regularity exponents for the process X.
Corollary 3.1. Let d ≤ 3, p ≥ q = 2m, m ∈ N+, β1 > 0, η < 1q. Assume
√
qkek, qk > 0, k ∈ N+, with eigenfunctions such that kekkE ≤ C, k∇ekk ≤ Cλ
1 2 k. Suppose that P k∈N+ qkλ 2β−1 k < ∞, for some 0 < β < 1 2. Then we have sup t∈[0,T ] kXN(t) − X(t)k Lq Lp(Ω)≤ C(T, p, q, X0)δt min(β,β12+η).
Proof. To prove that Lemma 3.1 holds true, it is sufficient to check the es-timate E[ sup
t∈[0,T ]
kω(t)kpE] ≤ C(T, p, Q) for p ≥ 1. This is a consequence of [10, Theorem 5.25].
It now remains to explain modifications concerning Lemma 3.4. More precisely, the control of the term I3 is modified as follows:
E[I3p] ≤ C(p)E h Z bscδt 0 S(s − r) − S(bscδt− r)dW (r) p Lq i + C(p)Eh Z s bscδt S(s − r)dW (r) p Lq i ≤ C Z bscδt 0 X k∈N+ (e−λk(s−r)− e−λk(bscδt−r))2q kdr p2 + C Z s bscδt X k∈N+ e−2λk(s−r)q kdr p2 ≤ C X k∈N+ λ−1k (1 − e−λk(s−bscδt))q k p2 + C X k∈N+ qk λk (1 − e−2λk(s−bscδt)) p 2 ≤ C(X k∈N+ qkλ2β−1k ) p 2δtβp.
Applying the same techniques as above concludes the proof of Corollary 3.1. Corollary 3.2. Let d = 1, β > 0 and p ≥ 2. If Q ∈ L02, X0∈ Hβ∩ E, then
there exists a constant C = C(X0, Q, T, p) such that
sup t∈[0,T ] kXN(t) − X(t)k Lp(Ω)≤ Cδt 1 2. If d = 2, 3, k(−A)12QkL0 2 < ∞, X0 ∈ H
β∩ E, then there exists a constant C0 =
C0(X0, Q, T, p) such that sup t∈[0,T ] kXN(t) − X(t)k Lp(Ω)≤ C 0δt1 2.
Proof. We first show the first assertion. The assumptions ensures that the method to obtain the strong convergence rates of the splitting scheme in the case Q = I is also available for the case Q ∈ L0
2. We only need to show that the a
priori estimate of ω, and I3 possess higher convergence speed than the case Q = I.
The Sobolev embedding theorem, the regularity result of stochastic convolution in [10, Theorem 5.15] and Burkerholder inequality yield that for p ≥ 2, there exsits
and E[I3p] ≤ C(p)E h Z bscδt 0 S(s − r) − S(bscδt− r)dWQ(r) pi + C(p)Eh Z s bscδt S(s − r)dWQ(r) pi ≤ C(p)Eh Z bscδt 0 (−A) −1 2(S(s − bscδt) − I) 2 (−A) 1 2S(bscδt− r)Q 2 L0 2 dr p 2i + C(p)Eh Z s bscδt S(s − r)Q 2 dr p 2i ≤ C(Q)δtp2.
The above properties, combined with the procedures in the proof of Theorem 3.1 shows the first assertion.
Denote Wγ= Rt 0(t−s) −γS(t−s)(−A)1 2dWQ(s). When d = 2, 3, k(−A) 1 2QkL0 2< ∞, Sobolev embedding theorem H1+2β ,→ E , 1
4 < β < 1
2, together with the the
fractional method and Lemma 3.2, yields that for p > 2, 1 4 < β < 1 2, 1 2 > γ > β + 1 p, E h sup s∈[0,T ] kω(s)kpEi≤ Eh sup s∈[0,T ] kω(s)kp H1+2β i ≤ CEh sup s∈[0,T ] kGγWγ(s)k p H2β i ≤ C Z T 0 E h kWγ(s)kpH2β i ds ≤ C Z T 0 s−2γkS(s)(−A)12Qk2 L0 2ds p2 ≤ C(T, Q, p). Combining with the continuity of stochastic convolution
E[I3p] ≤ C(p)E hZ bscδt 0 (−A) −1 2(S(s − bscδt) − I) 2 (−A) 1 2S(bscδt− r)Q 2 L0 2 dr p 2i + C(p)Eh Z s bscδt S(s − r)Q 2 dr p 2i ≤ C(Q)δtp2. The a priori estimate of E[ sup
t∈[0,T ]
kω(t)kpE] and some procedures in the proof of Theorem 3.1, we get the second assertion.
4. Higher strong convergence rate using exponential integrability properties (regular noise, dimension 1)
This section is devoted to two contributions. First, we investigate exponential integrability properties of the exact and numerical solutions X and XN, in
dimen-sion d = 1, 2, 3. We also derive useful a priori estimates in the H2 norm. This
requires additional regularity conditions on the operator Q, and the initial condi-tion X0: it is assumed that k(−A)
1 2QkL0
2 < ∞ and X0 ∈ H
2. Second, we prove
that the splitting scheme, Eq. (2), in the one-dimensional case d = 1, has a strong order of convergence equal to 1. Note that this higher order of convergence may be obtained since the stochastic convolution is not discretized. Up to our knowledge, this is the first proof that a temporal discretization scheme has a strong order of convergence equal to 1 for the stochastic Allen-Cahn equation.
Like in Section 3, it is assumed for simplicity that the initial condition X0 is
under appropriate assumptions: for instance, conditions of the type E[eckX0k2H1] < ∞ for some c < ∞ are required when studying exponential integrability properties. 4.1. A priori estimates and exponential integrability properties. In order to show an improved strong error estimate, with order 1 in some cases, we need to prove additional a priori estimates, and to study the exponential integrability properties for d = 1, 2, 3, in some well-chosen Banach spaces.
We first state the following result. The proof is standard, using Itˆo’s formula and the one-sided Lipschitz condition on F , and it is thus left to the interested readers.
Lemma 4.1. Assume that d ≤ 3, k(−A)12Qk
L0
2 < ∞ and X0∈ H
1. Let p ≥ 1.
Then the solution X of Eq. (1) satisfies the a priori estimates
E h sup t∈[0,T ] kX(t)k2+ Z T 0 kX(t)k2H1dt + Z T 0 kX(t)k4L4dt pi ≤ C(X0, T, Q, p) and E h sup t∈[0,T ] kX(t)k2 H1+ Z T 0 kX(t)k2 H2ds pi ≤ C(X0, T, Q, p).
To show the exponential integrability of X, we quote an exponential integra-bility lemma, see [8, Lemma 3.1], see also [6] for similar results.
Lemma 4.2. Let H be a Hilbert space, and let X be an H-valued adapted sto-chastic process with continuous sample paths, satisfying Xt= X0+R
t
0µ(Xr)dr +
Rt
0σ(Xr)dWrfor all t ∈ [0, T ], where almost surely
RT
0 (kµ(Xt)k+kσ(Xt)k
2)dt < ∞.
Assume that there exist two functionals U and U ∈ C2(H; R), and α ≥ 0, such
that for all t ∈ [0, T ]
DU (x)µ(x) +trD 2U (x)σ(x)σ∗(x) 2 + kσ∗(x)DU (x)k2 2eαt + U (x) ≤ αU (x). Then sup t∈[0,T ] E exp U (Xt) eαt + Z t 0 U (Xr) eαr dr ≤ eU (X0).
We are now in position to state a first exponential integrability result, which will be improved below. For the reader’s convenience, we omit standard truncations and regularization procedures.
Proposition 4.1. Let d ≤ 3, and assume that k(−A)12Qk
L0
2 < ∞ and X0 ∈ H1. Then for any ρ, ρ1> 0, there exist α = λ(ρ, Q) ∈ (0, ∞) and α1= λ(ρ1, Q) ∈
Proof. Define µ(x) = Ax − x3+ x, σ(x) = Q, U (x) = ρkxk2 and U1(x) =
ρ1k∇xk2. Then note that for ρ > 0,
hDU (x), µ(x)i +1 2tr[D 2U (x)σ(x)σ∗(x)] +1 2kσ(x) ∗DU (x)k2 = 2ρhx, Ax − x3+ xi + ρkQk2L0 2+ 2ρ 2kQ∗xk2 ≤ −2ρk∇xk2+ 2ρkxk2− 2ρkxk4L4+ ρkQk 2 L0 2+ 2ρ 2 kxk2kQk2L0 2 ≤ −2ρk∇xk2− 2ρkxk4 L4+ ρkQk 2 L0 2+ (2ρ + 2ρ 2kQk2 L0 2)kxk 2. Let α ≥ 2ρ + 2ρ2kQk2 L0 2 , and define U (x) = 2ρk∇xk2+ 2ρkxk4L4− ρkQk 2 L0 2. Then one may apply Lemma 4.2, which yields
E h expe−αtρkX(t)k2+ 2ρ Z t 0 e−αsk∇X(s)k2ds + 2ρ Z t 0 e−αskX(s)k4 L4ds i ≤ Ehe ρkQk2 L02 λ eρkX0k2 i ≤ eρkX0k2.
The second inequality is obtained with similar arguments and the fact that H1= H01: hDU1(x), µ(x)i + 1 2tr[D 2U 1(x)σ(x)σ∗(x)] + 1 2kσ(x) ∗DU 1(x)k2 ≤ −2ρ1hAx, Axi − 6ρ1h∇x, ∇xx2i + ρ1k∇Qk2L0 2+ (2ρ1+ 2ρ 2 1k∇Qk 2 L0 2)k∇xk 2.
It remains to apply Lemma 4.2, to get for α1≥ 2ρ1+ 2ρ21k∇Qk2L0 2 , E h expe−α1tρ 1kX(t)k2+ 2ρ1 Z t 0 e−α1skAX(s)k2dsi ≤ Ehe ρ1k∇Qk2L0 2 α1 eρ1k∇X0k2i≤ eρ1k∇X0k2.
This concludes the proof of Proposition 4.1. The use of Gagliardo–Nirenberg–Sobolev inequalities (see e.g. [24]) then al-lows us to improve the result of Proposition 4.1 as folal-lows: we control exponential moments of the type EhexpRT
0 ckX(s)k 2 Eds
i
with arbitrarily large parameter c ∈ (0, ∞). This result is crucial in the approach used below to obtain higher rates of convergence for the splitting scheme.
Proposition 4.2. Let d ≤ 3, and assume that k(−A)12Qk
L0
2 < ∞ and X0 ∈ H1. Then the solution X of (1) satisfies, for any c > 0,
E h exp Z T 0 ckX(s)k2Edsi≤ C(c, d, T, X0, Q) < ∞.
Thanks to Young’s inequality, for all ∈ (0, 1), there exists C1() ∈ (0, ∞) such that kXk2 E ≤ C1k∇Xk 2 3kXk 4 3 L4 ≤ k∇Xk2+ kXk4L4+ C1() .
Choose = (c) ≤ ceραT ≤ 1. Then, using Cauchy-Schwarz inequality, one gets
E h exp Z T 0 ckX(s)k2Edsi ≤ Ehexp Z T 0 ck∇Xk2+ ckXk4L4+ C1(, c) dsi ≤ eC1(,c)T s E h exp Z T 0 2ck∇Xk2dsi s E h exp Z T 0 2ckXk4 L4ds i ≤ C(c, 1, T, X0, Q),
thanks to Proposition 4.1, since 2c ≤ ceραT. This concludes the treament of the case d = 1.
When d = 2, resp. d = 3, we apply the Gagliardo-Nirenberg-Sobolev inequality, kXkE ≤ C2kAXk 1 3kXk 2 3 L4, resp. kXkE ≤ C3kAXk 3 5kXk 2 5
L4. In both cases, applying
Young’s inequality, for any ∈ (0, 1), there exists Cd() ∈ (0, ∞) such that
kXk2 E ≤ kAXk2+ kXk4L4+ Cd() . Choose = (c) ≤ min(ceραT, ρ1 eα1T) ≤ 1. Then E h exp Z t 0 ckX(s)k2Edsi ≤ Ehexp Z T 0 ckAXk2+ ckXk4L4+ Cd(, c) dsi ≤ C(, c, Cd, T ) s E h exp Z T 0 2ckAXk2dsi s E h exp Z T 0 2ckXk4 L4ds i ≤ C(c, Cd, T, X0, Q),
using Proposition 4.1, and the condition on .
This concludes the proof of Proposition 4.2. To conclude this section, we give an additional a priori estimate, with higher order spatial regularity for the solution X of Eq. (1).
Proposition 4.3. Let d ≤ 3, k(−A)12Qk
L0
2 < ∞ and X0 ∈ H
2. Then the
solution X ∈ H2, a.s. Moreover for any p ≥ 2,
sup s∈[0,T ]E h kX(s)kp H2 i ≤ C(T, Q, X0, p).
Proof. By the mild form of Y , we get
The boundedness of S(·) and the calculus inequality in the Sobolev spaces (see e.g. [23]) leads that kS(t − s)F (Y (s) + ω(s)))kH2 ≤ CkY (s) + ω(s)kH2+ kX(s)kH2kX(s)k2E+ kX(s)2kH2kX(s)kE ≤ CkY (s) + ω(s)kH2+ kX(s)kH2kX(s)k2E+ kX(s)k2 H2kX(s)kE .
Gronwall inequality, together with Sobolev embedding theorem, implies that
kY kH2≤ C exp(C Z T 0 kX(s)kEkX(s)kH2ds) sup t∈[0,T ] kS(t)X0k + Z T 0 (1 + kX(s)kEkX(s)kH2)kω(t)kH2dt .
Taking expectation, the exponential integrability in Proposition 4.2, and the regu-larity of the stochastic convolution,
sup t∈[0,T ] E h kAω(s)kpi≤ sup t∈[0,T ] E h kA Z t 0 S(t − s)dWQ(s)kpi ≤ C sup t∈[0,T ] E hZ t 0 k(−A)12S(t − s)(−A) 1 2Qk2 L0 2ds p2i ≤ C(T, Q, p), yields that E[kX(s)kpH2] ≤ CE[kω(s)k p H2] + C E h exp(2pC Z T 0 kX(s)kEkX(s)kH2ds) i12 × sup t∈[0,T ] kS(t)X0k2pH2+ E hZ T 0 kX(s)kEkX(s)kH2kω(s)kH2ds 2pi12 .
By Gagliardo–Nirenberg inequality in d = 1, 2, 3, and Young inequality, we get
kX(s)kEkX(s)kH2 ≤ CkX(s)kH2k∇X(s)k 1 2kX(s)k 1 2 L4 ≤ kX(s)k2H2+ k∇X(s)k 2+ kX(s)k4 L4+ C(), d = 1 kX(s)kEkX(s)kH2 ≤ CkX(s)k 4 3 H2kX(s)k 2 3 L4 ≤ kX(s)k2 H2+ kX(s)k 4 L4+ C(), d = 2 kX(s)kEkX(s)kH2 ≤ CkX(s)k 8 5 H2kX(s)k 2 5 L4 ≤ kX(s)k2 H2+ kX(s)k 4 L4+ C(), d = 3.
Combining with Proposition 4.1, we get the boundedness of this exponential mo-ment expCR0TkX(s)kEkX(s)kH2ds
. The estimation of EhR0TkX(s)kEkX(s)kH2kω(s)kH2ds
is similar. Gagliardo–Nirenberg–Sobolev inequality, together with Sobolev embed-ding L4,→ H1, yields that for d = 1,
E hZ T 0 kX(s)kEkX(s)kH2kω(s)kH2ds 2pi ≤ CEh( Z T 0 kX(s)k2 H2ds) 2pi + CEh Z T 0 kX(s)k4p H1kω(s)k 4p H2ds i , for d = 2, E hZ T 0 kX(s)kEkX(s)kH2kω(s)kH2ds 2pi ≤ CEh( Z T 0 kX(s)k2 H2ds) 2pi + CEh Z T 0 kX(s)k4p H1kω(s)k 6p H2ds i , for d = 3, E hZ T 0 kX(s)kEkX(s)kH2kω(s)kH2ds 2pi ≤ CEh( Z T 0 kX(s)k2 H2ds) 2pi + CEh Z T 0 kX(s)k4p H1kω(s)k 10p H2 ds i .
Combining the a priori estimate in Lemma 4.1 and the above inequalities, we finish
the proof.
4.2. Strong convergence order 1 of the splitting scheme. In this part, we focus on the sharp strong convergence rate of XN in d = 1. The main reason
why we could not obtain higher strong convergence rate in d = 2, 3 is that this splitting up strategy will destroy the exponential integrability in L4([0, T ]; L4) and
L2
([0, T ]; H2) of the original equation and that the a priori estimate of the auxiliary
process ZN
in H2
can not be obtained, since the Sobolev embedding E ,→ H1does
not hold. We also remark the a priori estimate in Lemma 4.3 holds for d = 2, 3. The study of higher strong convergence order for numerical schemes in high dimensional case will be studied in future works.
We first state the main result of this section.
Theorem 4.1. Assume that d = 1, k(−A)12QkL0
2 < ∞ and X0 ∈ H
2. The
proposed method possesses strong convergence order 1, i.e, for any p ≥ 1,
sup n≤NE h X(tn) − X N(t n) pi ≤ Cδtp.
To obtain the higher strong order of the splitting scheme, we consider the following auxiliary predictable right continuous process ZN such that ZN(t
n) =
XN(t
n), n ≤ N . The process ZN is defined by recursion. Let ZN(0) := X0and on
we rewrite the definition of ZN into an integration form, ZN(t) = ZN(tn−1) + Z t tn−1 F (ZN(s))ds, t ∈ [tn−1, tn), (14) ZN(tn) = S(δt)ZN(tn−1) + Z tn tn−1 S(δt)F (ZN(s))ds + Z tn tn−1 S(tn− s)dWQ(s). (15)
Letting n be n − 1 in the above equation and then plugging it into Eq. (14) yields that ZN(t) = S(δt)ZN(tn−2) + Z tn−1 tn−2 S(δt)F (ZN(s))ds + Z t tn−1 F (ZN(s))ds + Z tn−1 tn−2 S(tn−1− s)dWQ(s), t ∈ [tn−1, tn).
Repeating this process, we get, for t ∈ [tn−1, tn),
ZN(t) = S(tn−1)X0+ Z tn−1 0 S(tn−1− bscδt)F (ZN(s))ds + Z t tn−1 F (ZN(s))ds + Z tn−1 0 S(tn−1− s)dWQ(s), and ZN(tn) = S(tn)X(0) + Z tn 0 S(tn− bscδt)F (ZN(s))ds + Z tn 0 S(tn− s)dWQ(s).
4.2.1. A priori estimate for the auxiliary process. In order to get the strong convergence order, we also need the following a priori estimations of ZN.
Lemma 4.3. Assume that d = 1, k(−A)12Qk
L0
2 < ∞, kX0kH1 < ∞. Then for p ≥ 2, the auxiliary process ZN satisfies
E h sup s∈[0,T ] kZN(s)kp H1 i ≤ C(X0, p, T, Q).
Proof. We first show the estimation of sup
s∈[0,T ]
E[kZN(s)kpH1] ≤ C(T, p, Q, X0).
Since similar arguments in Lemma 3.1 implies that sup
s∈[0,T ]
E[kZN(s)kp] ≤ C(T, p, Q, X0),
it sufficient to show sup
s∈[0,T ]
E[k∇ZN(s)kp] ≤ C(T, p, Q, X0). For simplify the
pre-sentation, we only present the case p = 2. Consider the linear SPDE d bZ = A bZdt + dWQ(t) in local interval [tn−1, tn] with bZ(tn−1) = Φδt(ZN(tn−1)), we have
b
Z(tn) = ZN(tn). By Itˆo formula, we have
Then taking expectation yields that E[k∇ZN(tn)k2] ≤ E[k∇Φδt(ZN(tn−1))k2] + Z tn tn−1 k∇Qk2 L0 2ds. Since Φt−tn−1Z N(t
n−1) is the solution of d eZ = F ( eZ)dt with eZ(tn−1) = ZN(tn−1),
the similar arguments yields that
k∇Φt−tn−1(Z
N(t
n−1))k2≤ eCδtk∇ZN(tn−1)k2.
Combing the above estimations, we have for t ∈ [tn−1, tn),
E h k∇ZN(t)k2i≤ eCδtE h k∇ZN(tn−1)k2 i ≤ eCδteCδt E h k∇ZN(t n−2)k2 i + Cδt ≤ eCTkX 0k2+ C(Q, T ),
which implies that sup
s∈[0,T ]
E[k∇ZN(s)k2] ≤ C(T, 2, Q, X0). Similarly, we obtain the
uniformly boundedness of sup
s∈[0,T ] E h kZN(s)kp H1 i , p ≥ 2.
Now we are in position to show the desired result. By the argument in Lemma 3.1, we have E[ sup
n∈N
kX(tn)kpLq] ≤ C, q = 2m. Then we aim to prove that E[ sup
n∈N
k∇X(tn)kp] ≤
C. By the similar procedure of the previous proof of Lemma 3.1, we get
k∇XN(tn) − ω(tn) k2≤ (1 + δt) ∇ Φδt(XN(tn−1)) − Φδt(ω(tn−1)) 2 + Cδt(1 + kω(tn−1)k6H1).
Now, consider the SDEs d eZi = F ( eZi)dt with different inputs eZ1(tn−1) =
XN(t
n−1)) and eZ2(tn−1) = ω(tn−1), we get d( eZ1− eZ2) = (F ( eZ1) − F ( eZ2))dt
for t ∈ [tn−1, tn]. Further calculations, together with Gagliardo–Nirenberg, Holder
and Young inequalities, yield that
On the other hand, the monotonicity of F yields that the solution of d eZ = F ( eZ)dt satisfies for t ∈ [tn−1, tn],
sup
t∈[tn−1,tn]
k eZ(t)k2H1 ≤ eCδt(1 + k eZ(tn−1)k2H1).
The above inequality yields that
k∇Φt−tn−1(X N(t n−1)) − Φt−tn−1(ω(tn−1)) k2 ≤ eCδt(k∇(XN(t n−1) − ω(tn−1))k2+ eCδtδt(1 + kXN(tn−1)k6H1+ kω(tn−1)k6H1). Then discrete Gronwall’s inequality leads that
kXN(t n)k2H1 ≤ CkX0kH21+ Ckω(tn)k2H1+ C n−1 X j=0 δt(1 + kXN(tj)k6H1+ kω(tj)k6H1).
Taking expectation, we obtain for any p ≥ 2,
E h sup n≤N kXN(t n)k p H1 i ≤ C1 + kX0k p H1+ sup n≤NE h kXN(t n)k 3p H1 i + Ehsup n≤N kω(tn)k 3p H1 i . Denote Wγ = Rt 0(t − s) −γS(t − s)(−A)1
2dWQ(s). By the fractional method and Lemma 3.2, we have for β < 12, 12> γ > β + 3p1,
E h sup s∈[0,T ] kω(s)k3p H1 i ≤ Eh sup s∈[0,T ] kω(s)k3p H1+2β i ≤ CEh sup s∈[0,T ] kGγWγ(s)k 3p H2β i ≤ C Z T 0 E h kWγ(s)k 3p H2β i ds ≤ C Z T 0 s−2γkS(s)(−A)12Qk2 L0 2ds 3p2 ≤ C(T, Q, p), which implies that Ehsup
n≤N kXN(t n)k p H1 i
≤ C(T, X0, p, Q). Then the definition of
ZN yields that E h sup s∈[0,T ] kZN(s)kp H1 i ≤ C1 + Ehsup n≤N kXN(t n)k 3p H1 i ≤ C(T, X0, Q, p),
which completes the proof.
Similar to the procedures in the proof of Proposition 4.2, we show the following exponential integrability of ZN which is the key to get the higher strong convergence rate. The rigorous proof is that in each local interval, we first apply the truncated argument and the spectral Galerkin method, then use the Itˆo formula and Fatou lemma to get the evolution of Lyapunouv functions. For convenience, we omit these procedures.
Proposition 4.4. Assume that d = 1, k(−A)12Qk
L0
2 < ∞, kX0kH1 < ∞. Then we have for any c > 0,
Proof. In each subinterval [tn−1, tn], we define the process bZ as the solution of d bZ = A bZdt+dWQ(t), with bZ(t n−1) = ΦδtZN(tn−1). Denote µ(x) = Ax, σ(x) = Q, U (x) = ρkxk2 and U 1(x) = ρ1k∇xk2. we get for ρ, ρ1> 0, hDU (x), µ(x)i +1 2tr[D 2U (x)σ(x)σ∗(x)] +1 2kσ(x) ∗DU k2 = 2ρhx, Axi + ρkQk2L0 2+ 2ρ 2kQxk2 ≤ −2ρk∇xk2+ ρkQk2 L0 2+ 2ρ 2kQk2 L0 2kxk 2.
Lemma 4.2 yields that for α ≥ 2ρ2kQk2 L0 2 , E h expe−αtnρkZN(t n)k2 i ≤ eCδtE h expe−αtn−1ρkΦ δtZN(tn−1)k2 i . Since Φt−tn−1Z N(t
n−1) is the solution of d eZ = F ( eZ)dt with eZ(tn−1) = ZN(tn−1)
in [tn−1, tn], similar calculation, together with H¨older and Young inequality, yields
E h expe−αtn−1ρkΦ δtZN(tn−1)k2 i = Ehexpe−αtn−1ρkZN(t n−1)k2− e−αtn−12ρ Z tn tn−1 kZN(s)k4 L4ds + e−αtn−12ρ Z tn tn−1 kZN(s)k2dsi ≤ eCδt E h expeαtn−1ρkZN(t n−1)k2− e−αtn−1ρ Z tn tn−1 kZN(s)k4 L4ds i ≤ eCδt E h expe−αtn−1ρkZN(t n−1)k2 i .
Then repeating the above procedures,
E h expe−αtnρkZN(t n)k2 i ≤ eCδt E h expe−αtn−1ρkZN(t n−1)k2 i ≤ eCtneρkX0k2. For t ∈ [tn−1, tn), we similarly have
Next, we focus on the exponential integrability in H1. Since d bZ = A bZdt + dWQ(t) in [tn−1, tn], with bZ(tn−1) = ΦδtZN(tn−1), for ρ1> 0, we have
hDU1(x), µ(x)i + 1 2tr[D 2U 1(x)σ(x)σ∗(x)] + 1 2kσ(x) ∗DU 1(x)k2 = −2ρ1hAx, Axi + ρ1k∇Qk2L0 2+ 2ρ 2 1k∇Qk2L0 2k∇xk 2,
which yields that for α1≥ 2ρ21k∇Qk2L0 2 , E h expe−α1tnρ 1k∇ZN(tn)k2 i ≤ eCδtE h expe−α1tn−1ρ 1k∇ΦδtZN(tn−1)k2 i .
Then the fact that Φt−tn−1Z
N(t
n−1) is the solution of d eZ = F ( eZ)dt in [tn−1, tn],
with eZ(tn−1) = ZN(tn−1), yields that for α1≥ 2ρe1,ρe1= e
2ρ2 1k∇Qk2L02Tρ 1, E h expe−α1tn−1ρ 1k∇ΦδtZN(tn−1)k2+ Z tn tn−1 e−α1s 2ρ1h∇ZN(s), (ZN(s))2∇ZN(s)ids i ≤ eCδt E h expe−α1tn−1ρ 1k∇ZN(tn−1)k2 i
Repeating the above procedures and taking α1≥ max(2ρ21k∇Qk2L0 2 , 2e2ρ 2 1k∇Qk 2 L02Tρ 1), we obtain sup t∈[0,T ]E h expe−α1tρ 1k∇ZN(t)k2 i ≤ Ceρ1k∇X0k2.
Now, we are in position to show the desired result (16). Gagliardo–Nirenberg– Sobolev inequality kZNk E ≤ C1k∇ZNk 1 3kZNk 2 3
L4, together with H¨older and Young inequalities, implies that
E h exp Z T 0 ckZN(s)k2Edsi ≤ Ehexp Z T 0 1 21k∇Z N(s)k2+1 22kZ N(s)k4 L4+ C(1, 2, c))ds i ≤ C(T, 1, 2, c) s E h exp Z T 0 1k∇ZN(s)k2ds i s E h exp Z T 0 2kZN(s)k4L4ds i . Choosing 2≤ e−αTρ, we have s E h exp Z T 0 2kZN(s)k4L4ds i ≤ eCTeρ2kX0k2. Taking 1≤e −α1Tρ1
T , together with Jensen inequality, yields that
s E h exp Z T 0 1k∇ZN(s)k2ds i ≤ sup s∈[0,T ] r E h expT 1k∇ZN(s)k2 i ≤ eCTeρ2k∇X0k2. The above two estimations leads the desired result.
4.2.2. Strong convergence order 1 of the splitting scheme. After establish the a priori estimates and the exponential integrability of both the exact and numerical solutions, we are in position to give the other main result on the strong convergence rate of the splitting scheme.
Proof of Theorem 4.1. The mild representation of X (5) and XN (8) yields that kX(tn) − XN(tn)k ≤ n−1 X j=0 Z tj+1 tj S(tn− s)(F (X(s)) − F (ZN(s))ds + n−1 X j=0 Z tj+1 tj (S(tn− s) − S(tn− tj))F (ZN(s))ds := II1+ II2.
By the smoothing properties of S(t), H1is an algebra and |F (z)| ≤ C(1 + |z|3), for 0 < η < 1, II2 is treat as follows:
II2= Z tn 0 (−A)ηS(tn− s)(−A)−η(I − S(s − bscδt))F (ZN(s))ds ≤ Cδt12+η 1 + sup s∈[0,T ] Z N(s) 3 H1 Z tn 0 (−A) ηS(t n− s) ds ≤ Cδt12+η 1 + sup s∈[0,T ] Z N (s) 3 H1 .
For convenience, we introduce the mapping G such that F (z1)−F (z2) = G(z1, z2)(z1−
z2), z1, z2∈ R, where G(z1, z2) = −(z12+ z 2 2+ z1z2) + 1. II1 is decomposed as II1≤ n−1 X j=0 Z tj+1 tj S(tn− s)G(X(s), ZN(s))(X(tj) − ZN(tj))ds + n−1 X j=0 Z tj+1 tj S(tn− s)G(X(s), ZN(s))(X(s) − X(tj))ds + n−1 X j=0 Z tj+1 tj S(tn− s)G(X(s), ZN(s))(ZN(s) − ZN(tj))ds := II11+ II12+ II13.
Direct calculations, together with Sobolev embedding and Gagliardo–Nirenberg inequality, yields that
and II13≤ 2C n−1 X j=0 Z tj+1 tj 1 + kX(s)k2L6+ kZN(s)k2L6 Z N(s) − ZN(t j) L6ds ≤ 2C n−1 X j=0 Z tj+1 tj sup s∈[0,tn] 1 + kX(s)k2L6+ kZN(s)k2L6 Z s tj F (ZN(r))dr L6ds ≤ 2Cδt sup s∈[0,tn] 1 + kX(s)k4L6+ kZ N(s)k4 L6+ kZ N(s)k6 L18 ≤ 2Cδt sup s∈[0,tn] 1 + kX(s)k4H1+ kZN(s)k6H1 . For II12, we have II12≤ n−1 X j=0 Z tj+1 tj S(tn− s)G(X(tj), ZN(tj))(X(s) − X(tj))ds + n−1 X j=0 Z tj+1 tj S(tn− s) G(X(s), ZN(s)) − G(X(tj), ZN(s)) (X(s) − X(tj))ds + n−1 X j=0 Z tj+1 tj S(tn− s) G(X(tj), ZN(s)) − G(X(tj), ZN(tj)) (X(s) − X(tj))ds := II121+ II122+ II123.
Using the mild form of X(s) (5) and Sobolev embedding H1,→ E , we have
For the last term, taking expectation, together with the independence of increments of Wiener process, the adaptivity of X, Fubini theorem and Burkholder-Davis-Gundy inequality, yields that for p ≥ 2,
E h n−1 X j=0 Z tj+1 tj S(tn− s)G(X(tj), ZN(tj)) Z s tj S(s − r)dWQ(r)ds pi = Eh n−1 X j=0 Z tj+1 tj Z tj+1 r S(tn− s)G(X(tj), ZN(tj))S(s − r)dsdWQ(r) pi ≤ C(p)Eh n−1 X j=0 Z tj+1 tj Z tj+1 r S(tn− s)G(X(tj), ZN(tj))S(s − r)dsQ 2 L0 2 dr p 2i ≤ C(p)δtp n−1 X j=0 Z tj+1 tj kX(tj)k2Lp(Ω;L6)+ kZ N(t j)k2Lp(Ω;L6)+ 1 X k∈N+ Qek 2 H1 ds p 2 ≤ C(T, Q, X0, p)δtp.
The definition of G implies that G is symmetric and |G(z1, z2)−G(z1, z3)| ≤ |z1||z2−
z3| + |z2− z3||z2+ z3|. Based on this property, we estimate III122 and III123 as
III122+ III123 ≤ 2C n−1 X j=0 Z tj+1 tj kX(s) − X(tj)k2L6(kX(s)kL6+ kX(tj)kL6+ kZN(s)kL6)ds + 2C n−1 X j=0 Z tj+1 tj kX(s) − X(tj)kL6kZN(s) − ZN(tj)kL6(kZN(s)kL6+ kZN(tj)kL6+ kX(tj)kL6)ds.
The continuity of X, the right continuity of ZN and Sobolev embedding theorem
lead that for s ∈ [tj, tj+1), η < 1,
where the two stochastic convolution terms can be bounded by Sobolev embedding L6,→ H1 and similar estimations for I3in Corollary 3.2, and
kZN(s) − ZN(t j)kL6 ≤ k Z s tn−1 F (ZN(r))drkL6 ≤ Cδt sup r∈[0,T ] 1 + kZN(r)kH1+ kZN(r)k3 H1 .
The above estimations, together with Young and H¨older inequality, implies that
III122+ III123 ≤ 2Cδtmin(1,2η) sup s∈[0,tn] 1 + kX0k4H2+ kX(s)k12H1+ kZN(s)k12H1 + 2C sup s∈[0,tn] kZN(s)kH1+ kX(s)kH1 n−1X j=0 Z tj+1 tj Z s tj S(s − r)dWQ(r) 2 H1 + k Z tj 0 (S(s − r) − S(tj− r))dWQ(r)k2H1 ! ds.
Since kX(tn) − ZN(tn)k ≤ II11+ II2+ II13+ II121+ II122+ II123, the discrete
Gronwall’s inequality in [7, Lemma 2.6] yields that
kX(tn) − ZN(tn)k ≤ C exp 2 n−1 X j=0 Z tj+1 tj kX(s)k2 E+ kZN(s)k2Eds ×II2+ II13+ II121+ II122+ II123 .
4.2 and 4.4, we obtain for p ≥ 1, 12< η < 1, sup n≤NE h kX(tn) − XN(tn)kp i ≤ C(p) 14 s E h exp(4p Z T 0 kX(s)k2 E) i 1 4 s E h exp(4p Z T 0 kZN(s)k2 E) i q E[II22p] + q E[II132p] + q E[II1212p ] + p E[(II122+ II123)2p] ≤ Cδt(1 2+η)p s E h 1 + sup s∈[0,T ] Z N(s) 6p H1 i + Cδtp s E h sup s∈[0,T ] 1 + kX(s)k8p H1+ kZ N(s)k12p H1 i + Cδtp s E h sup s∈[0,T ] 1 + kX(s)k10p H1 + kZ N(s)k10p H1 i + Cδtp N −1 X j=0 δt r E h k(−A)X(tj)k2p(kX(tj)k 4p H1+ kZ N(t j)k 4p H1) i + Cδtmin(1,2η)p s E h 1 + kX0k8pH2+ sup s∈[0,T ] kX(s)k24p H1 + kZ N(s)k24p H1 i ≤ C(T, p, Q, X0)δtp 1 + N −1 X j=0 δt 14 r E h k(−A)X(tj)k4p i 1 2 r E h kX(tj)k 8p H1+ kZ N(t j)k 8p H1 i ≤ C(T, p, Q, X0)δtp,
which completes the proof.
As a direct consequence of the Theorem 4.1 above, we have the following stronger error estimation.
Corollary 4.1. Assume that d = 1, k(−A)12Qk
L0
2 < ∞ and X0∈ H
2. Then
for any p ≥ 1 and 0 < η < 1, sup n≤N kX(tn) − XN(tn)k Lp(Ω)≤ Cδt η .
Proof. Since for any q0 ≥ 1, based on Theorem 4.1, we have E h sup n≤N X(tn) − X N(t n) qi ≤ X n≤N E h X(tn) − X N(t n) qi ≤ Cδtq−1.
We completes the proof by taking 1 −1q0≥ η and q0≥ p.
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Univ Lyon, CNRS, Universit´e Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, F-69622 Villeurbanne, France
E-mail address: brehier@math.univ-lyon1.fr
1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China
E-mail address: jianbocui@lsec.cc.ac.cn(Corresponding author)
1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China
